# Are all abelian subgroups of symmetric groups generated by disjoint cycles?

That is, do all the subgroups of, say, S6, look sort of like the subgroup generated by the cycles (123) and (45)?

This seems like an overly simplistic characterization of the Abelian subgroups, but I can't think of any counterexamples.

• Would you call the subgroup of $S_4$ generated by the single element $(1\,2)(3\,4)$ call generated by disjoint cycles? – Hagen von Eitzen Feb 21 at 18:30

Consider the abelian subgroup $$\{\operatorname{id},(1\,2)(3\,4),(1\,2)(5\,6),(3\,4)(5\,6)\}$$ of $$S_6$$.

• Even "less disjoint": $\{\text{id},(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}$ in $S_4$. – Andreas Blass Feb 21 at 19:05

Particular examples were given. Here's a more general recipe.

What are transitive abelian subgroups of the group of permutations of a set $$X$$ (finite or not)? answer (exercise): these are the group of translations for some group structure on $$X$$.

On the other hand, a group generated by a family of disjoint cycles is not transitive, unless there's a single cycle (so the group is cyclic).

Hence for any abelian non-cyclic group $$G$$, the group $$G$$ is an abelian subgroup of $$\mathfrak{S}(G)$$, and is not generated by disjoint cycles. The first example is precisely the Klein 4-group as given by Andreas.

This is not the only obstruction, since Hagen's example (with 3 orbits of size 2) is contained, but not equal, to the group generated by disjoint cycles.